From the arithmetic sequence: a, a+c, a+2c.
From the geometric sequence: a, ad, ad^21.
a+c=ad, so d=(a+c)/a. a+2c=ad^21, so a((a+c)/a)^2(a+2c)=1, simplifying neatly to a=c^2, when d=1/c+1.
So the series are: c^2,c^2+c,c^2+2c, and c^2, c^2(1/c+1) =c^2+c, and c^2(1/c+1)^2 = (c+1)^2. These solve for all c, so Excel produces a chart of values, and it is just necessary to check for those compliant with the problem.
Looking into it a bit further though, we can derive two separate solution families from the data given:
A = 5(9n^2+6n+1), B = 5(30n^2+19n+3), c = 10(3n+1), and
A = 45n^2, B = 5(30n^2n), c = 30n,
(with a small 'c' in each case)
The first family gives for n= 0,1,1, and 2, c= 10,20,40,and 50, solutions 1,2,5,and 6:
n,c,A,B,C,(A+B+C), solution number
2 50 125 425 2600 3150 6
1 20 20 70 440 530 2
0 10 5 15 80 100 1
1 40 80 260 1520 1860 5
The second family gives for n=1, 1 c=30, and 30, solutions 3 and 4.
n,c,A,B,C,(A+B+C), solution number
1 30 45 155 960 1160 4
1 30 45 145 840 1030 3
On checking, these values agree with those given by Charlie's program.
Edited on March 24, 2016, 11:17 pm

Posted by broll
on 20160324 23:07:43 